\frac{dS}{dt} = -\delta_1 D + \delta_2 J - \delta_3 U
Where:
1,2,3>0 \delta_1, \delta_2, \delta_3 > 01,2,3>0represent sensitivities to degradation, justice perception, and material incentives.
Positive S predicts authoritarian drift (passive compliance), negative predicts anarchic tendencies (active resistance or unrest).
5. Feedback Loops and Coupled Dynamics
These differential equations are mutually coupled, forming recursive feedback loops:
Increases in U elevate Ar, which in turn increases Ap, reinforcing D.
Higher D reduces S, potentially destabilizing the system U and J affecting future and dynamics.
Cognitive dissonance (C) and social norms (Smeso) amplify non-linear effects, introducing bifurcation points where small changes in incentives or justice perception can dramatically shift systemic outcomes.
This coupled system provides the formal foundation for numerical simulations, sensitivity analysis, and scenario modeling, enabling predictions of critical thresholds for democratic stability under varying conditions of vote buying and perceived injustice.
C. Derivation of Equilibrium Points and Conditions for Bifurcation